The schema, data mappings, and queries can be exported with the Export Solution command on the GraphStudio home page. You can then use Import Solution to import to another machine.
Note the following however:
Export Solution does not include the database data. If you have the original data files used to populate your graph, you should use the same files again. When you look at the imported solution, on the Data Mapping page, it will show some file names, but those are just references; it does not have the actual files. You should add and upload the files again, resulting in duplicate file icons on the Data Mapping page. After you have loaded the data to the new server, you can delete the duplicate file icons.
We are working on a product update in which Export/Import will give you the option to include the data.
If you do not have the original files, you can manually write out the data by writing queries which select and output each vertex type and each edge type to a CSV file.